Question

the base of a solid is the region bounded by y=x²−3 and y=−2 . find the volume of the solid given that the cross sections perpendicular to the x -axis are squares.

Answer 1

The volume of the solid is
`\(_((125/3))\)` cubic units. This is determined by integrating the square of the difference between the upper and lower functions,
`\( (x^2 - (-2))^2 \)`, from the x-coordinate where the curves intersect to the limits of integration.

Step-by-step explanation:

To find the volume of the solid with cross sections perpendicular to the x-axis as squares, integrate the area of one such square from the x-coordinate where the curves
`\(y = x^2 - 3\) and \(y = -2\)` intersect. The height of the square is the difference between the upper and lower functions:
`\(x^2 - (-2)\)`. The area of the square is then squared, representing the volume of an infinitesimally thin slice perpendicular to the x-axis. Integrating this over the given interval yields the total volume.

In the integral,
`\(\int_{{x_1}}^{{x_2}} (x^2 - (-2))^2 \,dx\), \(x_1\) and \(x_2\)` are the x-coordinates of the intersection points. Evaluate this integral to get the volume.

The process involves squaring the height of the square to represent the volume at a given x-value, then integrating across the entire interval to accumulate these volumes. The limits of integration are determined by finding the points of intersection between the curves. This yields the final volume of
`\(_((125/3))\)` cubic units for the solid.

Answer 2

The volume of the solid formed between y = x² - 3 and y = -2 with square cross sections is found by integrating the area of the squares from x = -1 to x = 1. The area of each square is (1 - x²)².

Step-by-step explanation:

To find the volume of the solid with a base bounded by y = x² - 3 and y = -2, and with cross sections perpendicular to the x-axis being squares, we need to calculate the integral of the area of these cross sections over the interval for which the solid exists.

The first step is to determine the interval of x for which the solid is defined. This is found by solving for x when y = x² - 3 equals y = -2. Solving the equation x² - 3 = -2 gives x = ±1. Therefore, the solid exists on the interval [-1, 1].

The side length of each square cross section is given by the distance between the curves, which is (-2) - (x² - 3) = 1 - x². The area of each square is then (1 - x²)². To find the volume, we integrate the area of the cross-section from x = -1 to x = 1:

∫-11 (1 - x²)² dx

This integral gives us the total volume of the solid.